A study of stochastic programming having some continuous random variables

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1 Itratoal Joural of Egrg Trds ad Tchology (IJETT) Volu 7 Nur 5 - July 06 A study of stochastc prograg havg so cotuous rado varals Mr.Hr S. Dosh, Dr.Chrag J. Trvd, Assocat Profssor, H Collg of Corc Navragura, Ahdaad Had & Assocat Profssor R J T Corc Collg,Vastrapur, Ahdaad ASTRACT I s papr, w hav statd soluto of stochastc prograg prol whr coffcts octv fucto follow ufor dstruto ad so of rght had sd paratrs follow varyg cotuous proalty dstrutos w kow a ad varac.to gt soluto of statd prol, w frst covrt stochastc prol to a quvalt dtrstc prol. Th a stadard lar/o-lar od s appld to solv trasford odl. Hypotcal apl has gv to llustrat odl. Kywords Ufor dstruto, potal dstruto, oral dstruto, lar/o-lar prograg od.. INTRODUCTION I practc, stochastc prograg s a procdur of dtrg optu soluto to costrad prols... s tchqu so of charactrstcs udr study ar ucrta ad hc ar rado varals possssg kow proalty dstruto. I ral lf stuatos, o has to dal w dcso akg prols varous flds, lk - aagt scc, dcal scc, socologcal aalyss, vrotal scc, grg ad tchology whch grally tchqus of stochastc prograg ar usd.a stochastc prograg odl s forulatd wh optal dcso s ucrta to happg of futur vts, Furr, optu soluto ca drvd aftr followg study of radoss of phoo. Th frst prso Datzg [4] has troducd stochastc prograg prols.susqutly, al [], Datzg ad Madasky [5] has was furr dvlopd s odl.acoparal cov progra of stochastc prograg udr ucrtaty was dvlopd y Wts [0]. A lographcal study of approatly 5 rsarch paprs durg was pulshd y Va dr Vlrk [9].A two- stag stochastc quadratc prograg odl w fd rsourcs was studd y Khor t al. [6].A two- stag stochastc tgr prograg odl for tgratd optzato of powr producto ad tradg whch clud a spcfc asur accoutg for rsk aagtwas proposd y rald t al.[]. A two- stag stochastc prograg odl for chcal producto plag optzato w aagt of varous ucrtats has studd y t al. [7], h has appld Mot Carlo sulato od. A soluto procdur for solvg two- stag stochastc lar prograg prol cosdrg o radoss ad trval paratrs prol forulato was stalshd y ark t al. []. Th da of s papr has otad fro papr pulshd y S.K ark, M.P wal ad D. Chakravarty 0 [8]. Th ttl of papr was Two stag stochastc prograg prols volvg so cotuous rado varals. Followg aov ltraturs studd y varous rsarchrs, whch spr us to troduc a w soluto procdur for solvg stochastc prograg prol whr coffcts octv fucto ar rado varals whch follows ufor dstruto ad so of rght had sd paratrs follow r ufor or potal or oral dstruto w kow a ad varac. Th soluto procdur has llustratd w sutal urcal apls.. STOCHASTIC PROGRAMMING MODE t us cosdrd as stochastc prograg odl havg so rado paratrs whch s to optzd. A aatcal for of a stochastc prograg odl ca dfd as udr: Suct to M Z= c...[] a,,..., r h s,,..., l...[] s s 0,,,...,,,,..., ar dcso varals, a,,...,,,,..., ar ISSN: Pag 86

2 Itratoal Joural of Egrg Trds ad Tchology (IJETT) Volu 7 Nur 5 - July 06 coffcts of tchologcal atr, c,,,..., ar rado varals assocatd w octv fucto whch follows ufor dstruto ovr spcfd rag A to,,,..., ad so of rght had sd paratrs,,,..., ar cosdrd as rado varals whch follow r ufor, or potal, or oral dstrutos w dft a ad varac.. Maatcal Modl: Maatcally stochastc prograg prol s wrtt as udr: M Z = E( c ) pe y...[] Suct to y a,,..., r h s,,..., l...[4] s s y 0, 0,,,...,,,,..., t s assud at dcso varals,,,..., ad y,,,..., ar dtrstc prol, p,,,..., ar palty costs assocatd w dscrpacy y,,,..., ad E rprsts pctd valu assocatd w rado varals c,,,..., ad,,,..., Now t s assud at c,,,..., ar rado varals follow ufor dstruto ovr rag A to,,,...,. Hc proalty dsty fucto of c s gv as f ( c ), A c,,..., A 0, orws. A W a Ec ( )..[5] ( A) Var( c ),, [6]. ad Thrfor, aov statd odl [] ad [4] ca rducd to Suct to A p E y M Z =...[7] y a,,..., r h s,,..., l...[8] s s y 0, 0,,,...,,,,..., Th dtrstc for aov stochastc prol wh rght had sd paratrs,,,..., follow so cotuous proalty dstruto ca otad as udr:... Cas :,,,..., follows ufor dstruto. Now t s assud at,,,..., ar dpdt ufor rado varals w U W a E ( )...[9] ad ( U ) Var( ),, [0] Th proalty dsty fucto of ufor rado varal,,,..., s gv as f ( ),,,..., U 0, orws. ad U valus of,,,..., Now, ufor rado varal U u ad au ufor rado varal E y E g whr g a, g 0,,..., g d,,..., U E U g U y ( ),,...,...[] U ISSN: Pag 87

3 Itratoal Joural of Egrg Trds ad Tchology (IJETT) Volu 7 Nur 5 - July 06 Usg rsult [] stochastc prograg odl [7] ad [8] dtrstc odl ca wrtt as M Z = Suct to A p U g U U r h s,,..., l s s 0,,,..., g a,,..., ( )...[]...[] Th soluto of aov prol ca otad y usg tchqus of lar prograg. follows... Cas :,,,..., Epotal dstruto. Now t s assud at,,,..., ar dpdt potal rado varals w W a E ( )...[4] ad Var( ),,......[5] Th proalty dsty fucto of potal rado varal,,,..., gv as s f ( ), 0,,,..., 0,, ohrws 0 s rat paratr of dstruto Now, ufor rado varal E y E g whr g a, g 0,,..., E y g d,,..., 0 g E y g,,...,...[6] Usg rsult [6] stochastc prograg odl [7] & [8] dtrstc odl ca wrtt as M Z = Suct to A g p g r h s,,..., l s s 0,,,..., g a,,...,...[8]...[7] Th soluto of aov prol ca otad y usg tchqus of o-lar prograg algor.... Cas :,,,..., follows Noral dstruto. Now t s assud at ar dpdt oral rado varals w a E ( )...[9] ad. Var ( ),, [0] Th proalty dsty fucto of rado varal,,,..., s gv as oral f ( ),, 0,,,..., 0,, ohrws ad ar paratrs of dstruto Now, oral rado varal E y E g whr g a, g 0,,..., E y g d,,..., E y g ( g ) g,,..., g...[] ISSN: Pag 88

4 Itratoal Joural of Egrg Trds ad Tchology (IJETT) Volu 7 Nur 5 - July 06 Usg rsult [] stochastc prograg odl [7] & [8] dtrstc odl ca wrtt as A M Z =...[] g g p g ( g ) Suct to r h s,,..., l s s 0,,,..., g a,,...,...[] Th soluto of aov prol ca otad y usg tchqus of o-lar prograg algor.. Hypotcal Eapl: Solv followg stochastc prograg odls w ut palty cost cosdrg all c as ufor rado varals ad so of rght had sd paratrs follow ufor, potal ad oral rado varals It s gv at c s a ufor rado varal ovr rag 0 to 0, c s a ufor rado varal ovr rag 40 to 80,ad c s a ufor rado varal ovr rag 5 to 45. Also E( ) 5, E( ) 0, E( ) 4 ad V ( ) 5, V ( ) 6, V ( ) 6. Th prol s to Suct to M z c c c ,,,. Soluto Th rado varals c, c ad c follows ufor dstruto so y ar rplacd y r pctd valus gv low A E( c ),,, E( c ) 0, E( c ) 60, E( c ) 40, Hc gv prol ca rwrtt as udr: Th prol s to M z E g E g E g Suct to ,,,. whr, g 4 6 4, g 0, g 5 5 Cas : Suppos,, follow ufor dstruto usg forula of a ad varac au ad u valu of,, ar calculatd as udr: 6.4, U.66, 9.6, U 0.4, 7.07, U 0.9. Thus usg u ad au valus of,, dsrd quvalt dtrstc odl ca stalshd as udr M z g 0.9g.00g 6.84 Suct to ,,,. whr, g 4 6 4, g 0, g 5 5 Usg lar algors soluto of aov prol ca otad as udr:.68, 0.60, 0.00, z 99.0 Cas : Suppos,, follow potal dstruto usg forula of a ad ISSN: Pag 89

5 Itratoal Joural of Egrg Trds ad Tchology (IJETT) Volu 7 Nur 5 - July 06 varac valus paratr,, ca otad as udr: 0., 0.7, 0.4 Thus usg u ad au valus of,, dsrd quvalt dtrstc odl ca stalshd as udr M z g 0.005g 0.07g 40 8 g g g 49 Suct to ,,,. whr, g 4 6 4, g 0, g 5 5 Usg o-lar algors soluto of aov prol ca otad as udr:.00, 0.0,., z 5.04 Cas : Suppos,, follow oral dstruto fro gv as ad varacs 5, 0, 4, ad 5, 6, 6 Thus usg valus of paratr dsrd quvalt dtrstc odl ca stalshd as udr M z g 5 g 0 (g 0) (g 40) 5 6 (g 8) g g 0 g g g g 49 Suct to g ,,,. whr, g 4 6 4, g 0, g 5 5 Usg o-lar algors soluto of aov prol ca otad as udr:.00, 0.0,., z 0.58 srvcs of a atv laguag spakg collagu or assocat to rvw ad dt your susso. 4. CONCUSION I s papr w proposd a soluto procdur of stochastc prograg odl whch all coffcts of octv fucto follow cotuous ufor dstruto ad so of rght had sd paratrs follow r ufor or potal or oral dstruto w spcfd a ad varac. Th drvd procdur s succssfully vrfd y hypotcal urc apl. REFERENCES [] ark S.K., M.P. swal ad Chakravarty, Two-Stag stochastc prograg prols volvg trval dscrt rado varals, OPSEARCH, Vol.,49, pp 80-98,0. [] al, E.M., o zg a cov fucto suct to lar qualts, oural of royal statstcal socty, vol.7, pp.7-84, 955. [] rald, P., D. cofort, ad A. vol, a two- stag stochastc prograg odl for lctrc rgy producrs, coputrs & opratos rsarch, vol.5 pp.60-70, 008. [4] Datzg, G.., lar prograg udr ucrtaty, aagt scc, vol., pp.97-06, 955. [5] Datzg, G.., ac A. adasky, o soluto of two- stag lar progras udr ucrtaty, procdgs of 4 rkly syposu of aatcal statstcs ad proalty, uvrsty Calfora prss, rkly, vol., pp.65-76, 96. [6] Khor, C.S., A. lkal, ad P.l. douglas, stochastc rfry plag w rsk aagt, ptrolu scc ad tchology, vol.6, pp , 008. [7], Q.G., ad G.H. Haug, A act two- stag stochastc rgy systs plag odl for aagg gr housga sso at a ucpal lvl, rgy, vol.5, pp.70-80, 00. [8] S. K. ark, M.P.swal,D. Chakravarty, Two Stag Stochastc Prograg Prols Ivolvg So Cotuous Rado Varal, Vol. 7, No. 4, PP , 0. A. N. Ntraval ad. G. Haskll, Dgtal Pcturs, d d., Plu Prss: Nw York, 995, pp [9] Va dr Vlrk, M.H., stochastc prograg lography, World Wd W, ally.co.rug.l/sp.htl, [0] Wts, R.J.., prograg udr ucrtaty: quvalt cov progra: quvalt cov progra, SIAM oural o appld aatcs, vol.4, pp.89-05, 966. ISSN: Pag 90